A Computational Model for q-Bernstein Quasi-Minimal Bezier Surface

Daud Ahmad, M. Khalid Mahmood, Qin Xin, Ferdous M. O. Tawfiq, Sadia Bashir, Arsha Khalid

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A computational model is presented to ­nd the q-Bernstein quasi-minimal Bezier surfaces as the extremal of Dirichlet functional,
and the Bezier surfaces are used quite frequently in the literature of computer science for computer graphics and the related
disciplines. The recent work [1–5] on q-Bernstein–Bezier surfaces leads the way to the new generalizations of ´ q-Bernstein
polynomial Bezier surfaces for the related Plateau–Bézier problem. The q-Bernstein polynomial-based Plateau–Bézier problem is
the minimal area surface amongst all the q-Bernstein polynomial-based Bézier surfaces, spanned by the prescribed boundary. ´
Instead of usual area functional that depends on square root of its integrand, we choose the Dirichlet functional. Related
Euler–Lagrange equation is a partial differential equation, for which solutions are known for a few special cases to obtain the
corresponding minimal surface. Instead of solving the partial differential equation, we can fi­nd the optimal conditions for which
the surface is the extremal of the Dirichlet functional. We workout the minimal Bézier surface based on the q-Bernstein
polynomials as the extremal of Dirichlet functional by determining the vanishing condition for the gradient of the Dirichlet
functional for prescribed boundary. The vanishing condition is reduced to a system of algebraic constraints, which can then be
solved for unknown control points in terms of known boundary control points. The resulting Bézier surface is q-Bernstein–Bézier
minimal surface
Original languageEnglish
Article number8994112
Number of pages21
JournalJournal of Mathematics
Publication statusPublished - 22 Sept 2022


  • Bezier surfaces
  • q-Bernstein polynomial-based Plateau–Bezier problem
  • Dirichlet functional
  • q-Bernstein–Bezier minimal surface
  • optimization theory


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